mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, given two submanifolds ''A'' and ''B'' of a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

''X'' intersecting in two points ''p'' and ''q'', a Whitney disc is a mapping from the two-dimensional disc ''D'', with two marked points, to ''X'', such that the two marked points go to ''p'' and ''q'', one boundary arc of ''D'' goes to ''A'' and the other to ''B''.. Their existence and

embeddedness In economics and economic sociology, embeddedness refers to the degree to which economic activity is constrained by non-economic institutions. The term was created by economic historian Karl Polanyi as part of his substantivist approach. Polanyi ...

is crucial in proving the

cobordism theorem In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : ...

, where it is used to cancel the intersection points; and its failure in low dimensions corresponds to not being able to embed a Whitney disc. Casson handles are an important technical tool for constructing the embedded Whitney disc relevant to many results on topological four-manifolds.

Pseudoholomorphic In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or ''J''-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 b ...

Whitney discs are counted by the differential in

Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

intersection Floer homology.

References

{{topology-stub Geometric topology